If it's got a US flag hanging on it, is being driven by a Skipper with a US-issued license, or has a manifested cargo destined for a US port and/or owned by a US entity....then the US Coast Guard has jurisdiction ANYWHERE in the friggin' world!
The Cap of the LST knew he was risking his Ticket, but didn't really give a damn. What the Coasties had their panties most in a wad over was the age and lack of current sea experience in the crew, and that they were, IIRC, attempting to sail without a licensed Chief Engineer. The general fitness of a US-licensed crew is also the Coast Guard's responsibility.
Keep in mind....regardless of what you see on "Deadliest Catch", the Coasties are basically highly bureaucratized cops, not soldiers or sailors, and just LOVE sticking their noses into other peoples' business looking for revenue-generating "rule-breaking" that they can then write-up to justify their phony-baloney jobs...whether it makes a lick of sense or not!